Mass-preserving Spatio-temporal adaptive PINN for Cahn-Hilliard equations with strong nonlinearity and singularity

As one kind important phase field equations, Cahn-Hilliard equations contain spatial high order derivatives, strong nonlinearities, and even singularities. When using the physics informed neural network (PINN) to simulate the long time evolution, it is necessary to decompose the time domain to capture the transition of solutions in different time. Moreover, the baseline PINN can't maintain the mass conservation property for the equations. We propose a mass-preserving spatio-temporal adaptive PINN. This method adaptively dividing the time domain according to the rate of energy decrease, and solves the Cahn-Hilliard equation in each time step using an independent neural network. To improve the prediction accuracy, spatial adaptive sampling is employed in the subdomain to select points with large residual value and add them to the training samples. Additionally, a mass constraint is added to the loss function to compensate the mass degradation problem of the PINN method in solving the Cahn-Hilliard equations. The mass-preserving spatio-temporal adaptive PINN is employed to solve a series of numerical examples. These include the Cahn-Hilliard equations with different bulk potentials, the three dimensional Cahn-Hilliard equation with singularities, and the set of Cahn-Hilliard equations. The numerical results demonstrate the effectiveness of the proposed algorithm.